# Properties

 Label 77.2.m.a Level 77 Weight 2 Character orbit 77.m Analytic conductor 0.615 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 77.m (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{15})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{15}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{2} + ( -\zeta_{15} - \zeta_{15}^{6} ) q^{3} + ( -\zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{6} ) q^{4} + ( -1 + \zeta_{15} - \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{5} + ( 1 + \zeta_{15}^{3} + \zeta_{15}^{6} ) q^{6} + ( -2 + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - \zeta_{15}^{4} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{7} + ( 2 - 2 \zeta_{15}^{2} + \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{8} -2 \zeta_{15}^{7} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{15}^{4} - \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{2} + ( -\zeta_{15} - \zeta_{15}^{6} ) q^{3} + ( -\zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{6} ) q^{4} + ( -1 + \zeta_{15} - \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{5} + ( 1 + \zeta_{15}^{3} + \zeta_{15}^{6} ) q^{6} + ( -2 + 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - \zeta_{15}^{4} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{7} + ( 2 - 2 \zeta_{15}^{2} + \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{8} -2 \zeta_{15}^{7} q^{9} + ( -1 + \zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} + 2 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{10} + ( 2 - 4 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} - \zeta_{15}^{5} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{11} + ( 1 - \zeta_{15} - \zeta_{15}^{4} + \zeta_{15}^{5} ) q^{12} + ( \zeta_{15}^{2} + 3 \zeta_{15}^{3} + 3 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{13} + ( 2 + \zeta_{15}^{2} + 3 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{14} + ( -1 - \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{15} + ( 3 \zeta_{15}^{4} + 3 \zeta_{15}^{7} ) q^{16} + ( -2 - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{17} + ( -2 + 2 \zeta_{15} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + 2 \zeta_{15}^{7} ) q^{18} + ( -3 + 6 \zeta_{15} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{7} ) q^{19} + ( -3 \zeta_{15}^{2} + \zeta_{15}^{3} + \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{20} + ( -1 + 2 \zeta_{15}^{5} ) q^{21} + ( 2 \zeta_{15}^{2} + 3 \zeta_{15}^{3} + \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{22} + ( 5 - 2 \zeta_{15} - 2 \zeta_{15}^{4} + 5 \zeta_{15}^{5} ) q^{23} + ( -2 + \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{24} + ( -4 \zeta_{15} - 7 \zeta_{15}^{4} - 4 \zeta_{15}^{7} ) q^{26} -5 \zeta_{15}^{3} q^{27} + ( -1 - 2 \zeta_{15} + 2 \zeta_{15}^{5} - 3 \zeta_{15}^{6} ) q^{28} + ( 6 - 3 \zeta_{15}^{2} + 6 \zeta_{15}^{3} + 3 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{29} + ( 1 + 2 \zeta_{15} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{30} + ( -1 + 6 \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - \zeta_{15}^{5} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{31} + ( -1 + \zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - 5 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{32} + ( -4 + \zeta_{15} + 2 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{7} ) q^{33} + ( 4 - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{7} ) q^{34} + ( -6 + \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{5} - 2 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{35} + ( 2 - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{36} + ( -5 \zeta_{15} - 3 \zeta_{15}^{4} - 5 \zeta_{15}^{7} ) q^{37} + ( -3 \zeta_{15} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{6} ) q^{38} + ( 4 - \zeta_{15} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + \zeta_{15}^{5} + 3 \zeta_{15}^{6} - 4 \zeta_{15}^{7} ) q^{39} -5 \zeta_{15}^{4} q^{40} + ( -1 + \zeta_{15}^{2} - 4 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{41} + ( 1 - 2 \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{5} - 2 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{42} + ( 2 + 3 \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{7} ) q^{43} + ( -4 + 3 \zeta_{15} - 4 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{44} + ( 4 - 4 \zeta_{15} - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{4} + 2 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{45} + ( -3 + 2 \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{46} + ( 4 - 6 \zeta_{15} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{47} + ( 3 + 3 \zeta_{15}^{3} ) q^{48} + ( -5 + 5 \zeta_{15} + 3 \zeta_{15}^{3} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{5} + 5 \zeta_{15}^{7} ) q^{49} + ( 2 \zeta_{15} + 2 \zeta_{15}^{4} + 2 \zeta_{15}^{7} ) q^{51} + ( 2 \zeta_{15} + \zeta_{15}^{2} + \zeta_{15}^{5} + 2 \zeta_{15}^{6} ) q^{52} + ( 5 - 2 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 5 \zeta_{15}^{4} + 5 \zeta_{15}^{5} + 5 \zeta_{15}^{6} - 5 \zeta_{15}^{7} ) q^{53} + ( 5 \zeta_{15} + 5 \zeta_{15}^{4} ) q^{54} + ( -4 + 6 \zeta_{15}^{2} - 9 \zeta_{15}^{3} - 6 \zeta_{15}^{6} + 6 \zeta_{15}^{7} ) q^{55} + ( -1 - 2 \zeta_{15} + 4 \zeta_{15}^{2} - 4 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + \zeta_{15}^{5} + 4 \zeta_{15}^{7} ) q^{56} + ( -6 \zeta_{15}^{2} + 3 \zeta_{15}^{3} + 3 \zeta_{15}^{6} - 6 \zeta_{15}^{7} ) q^{57} + ( -6 - 3 \zeta_{15} - 6 \zeta_{15}^{5} + 6 \zeta_{15}^{7} ) q^{58} + ( -6 + 6 \zeta_{15} - 6 \zeta_{15}^{3} + 6 \zeta_{15}^{4} + 6 \zeta_{15}^{7} ) q^{59} + ( -2 + 3 \zeta_{15} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 3 \zeta_{15}^{5} + \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{60} + ( 2 - 2 \zeta_{15} + 6 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 6 \zeta_{15}^{7} ) q^{61} + ( -4 - \zeta_{15}^{2} - 4 \zeta_{15}^{3} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{62} + ( -6 \zeta_{15} - 2 \zeta_{15}^{6} ) q^{63} + ( -\zeta_{15}^{2} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{64} + ( 5 + 5 \zeta_{15} + 5 \zeta_{15}^{4} + 5 \zeta_{15}^{5} ) q^{65} + ( 5 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 5 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 3 \zeta_{15}^{6} - 5 \zeta_{15}^{7} ) q^{66} + ( 1 - \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + 10 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{67} + 2 \zeta_{15}^{7} q^{68} + ( -2 + 2 \zeta_{15}^{2} - 5 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{69} + ( 9 - 2 \zeta_{15} - 6 \zeta_{15}^{2} + 9 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 6 \zeta_{15}^{6} - 8 \zeta_{15}^{7} ) q^{70} + ( -2 + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{6} ) q^{71} + ( -2 + 2 \zeta_{15} - 2 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{72} + ( 9 - 3 \zeta_{15} - 6 \zeta_{15}^{2} + 9 \zeta_{15}^{3} - 9 \zeta_{15}^{4} + 6 \zeta_{15}^{6} - 9 \zeta_{15}^{7} ) q^{73} + ( 5 \zeta_{15} + 3 \zeta_{15}^{2} + 3 \zeta_{15}^{5} + 5 \zeta_{15}^{6} ) q^{74} + ( -3 - 6 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{7} ) q^{76} + ( -7 + 8 \zeta_{15} + 3 \zeta_{15}^{2} - 9 \zeta_{15}^{3} + 6 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 3 \zeta_{15}^{6} + 7 \zeta_{15}^{7} ) q^{77} + ( -7 + 4 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 4 \zeta_{15}^{7} ) q^{78} + ( 6 - 6 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - 9 \zeta_{15}^{7} ) q^{79} + ( 6 \zeta_{15} + 3 \zeta_{15}^{2} + 3 \zeta_{15}^{5} + 6 \zeta_{15}^{6} ) q^{80} + ( 1 - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{81} + ( 1 - \zeta_{15} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} + 2 \zeta_{15}^{7} ) q^{82} -9 \zeta_{15}^{3} q^{83} + ( 3 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{84} + ( -2 + 2 \zeta_{15}^{2} + 4 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{85} + ( -2 + 2 \zeta_{15}^{4} - 2 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{86} + ( 3 - 3 \zeta_{15} - 3 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{87} + ( 2 - 5 \zeta_{15} - 6 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{5} - 3 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{88} + ( -8 + 4 \zeta_{15} + 4 \zeta_{15}^{4} - 8 \zeta_{15}^{5} ) q^{89} + ( 4 \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{90} + ( 9 + \zeta_{15} - 6 \zeta_{15}^{2} + 3 \zeta_{15}^{5} - 2 \zeta_{15}^{6} - 9 \zeta_{15}^{7} ) q^{91} + ( -2 + 7 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - 7 \zeta_{15}^{6} + 7 \zeta_{15}^{7} ) q^{92} + ( 6 - 6 \zeta_{15} - 5 \zeta_{15}^{4} + 6 \zeta_{15}^{5} - 5 \zeta_{15}^{7} ) q^{93} + ( -2 + 4 \zeta_{15} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{94} + ( -3 - 9 \zeta_{15} + 12 \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 12 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{95} + ( 1 - 5 \zeta_{15} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{96} + ( -\zeta_{15}^{2} + 3 \zeta_{15}^{3} + 3 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{97} + ( 5 - 8 \zeta_{15} - 5 \zeta_{15}^{2} + 5 \zeta_{15}^{3} - 8 \zeta_{15}^{4} - 5 \zeta_{15}^{7} ) q^{98} + ( -4 + 2 \zeta_{15}^{2} - 4 \zeta_{15}^{3} - 8 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} + q^{3} + 2q^{4} - 5q^{5} + 4q^{6} - 5q^{7} + 10q^{8} - 2q^{9} + O(q^{10})$$ $$8q - 2q^{2} + q^{3} + 2q^{4} - 5q^{5} + 4q^{6} - 5q^{7} + 10q^{8} - 2q^{9} - 10q^{10} + 4q^{11} + 2q^{12} - 10q^{13} + 3q^{14} - 10q^{15} + 6q^{16} - 4q^{17} - 6q^{18} - 3q^{19} - 10q^{20} - 16q^{21} - 4q^{22} + 16q^{23} + 5q^{24} - 15q^{26} + 10q^{27} - 12q^{28} + 24q^{29} + 5q^{30} + 8q^{31} + 18q^{32} - 11q^{33} + 24q^{34} - 5q^{35} + 8q^{36} - 13q^{37} - 9q^{38} + 5q^{39} - 5q^{40} + 2q^{41} + 10q^{42} + 28q^{43} - 12q^{44} + 8q^{46} + 6q^{47} + 18q^{48} - 11q^{49} + 6q^{51} - 5q^{52} - 12q^{53} + 10q^{54} + 10q^{55} - 24q^{57} - 21q^{58} - 18q^{59} + 5q^{60} + 18q^{61} - 28q^{62} - 2q^{63} + 6q^{64} + 30q^{65} + 2q^{66} - 38q^{67} + 2q^{68} - 2q^{69} + 20q^{70} - 16q^{71} - 10q^{72} + 15q^{73} - 14q^{74} - 48q^{76} - 4q^{77} - 40q^{78} + 9q^{79} - 15q^{80} + q^{81} + 7q^{82} + 18q^{83} + 2q^{84} - 20q^{85} - 7q^{86} + 18q^{87} - 5q^{88} - 24q^{89} + 50q^{91} + 16q^{92} + 8q^{93} + 8q^{94} + 15q^{95} - 2q^{96} - 14q^{97} + 4q^{98} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1 - \zeta_{15}^{5}$$ $$-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −0.978148 + 0.207912i −0.104528 − 0.994522i 0.913545 − 0.406737i 0.669131 − 0.743145i 0.669131 + 0.743145i 0.913545 + 0.406737i −0.978148 − 0.207912i −0.104528 + 0.994522i
0.0646021 0.614648i 0.669131 + 0.743145i 1.58268 + 0.336408i −2.04275 0.909491i 0.500000 0.363271i −0.0510966 + 2.64526i 0.690983 2.12663i 0.209057 1.98904i −0.690983 + 1.19682i
9.1 1.08268 + 1.20243i 0.913545 + 0.406737i −0.0646021 + 0.614648i −2.18720 0.464905i 0.500000 + 1.53884i −2.53158 0.768834i 1.80902 1.31433i −1.33826 1.48629i −1.80902 3.13331i
16.1 −1.58268 0.336408i −0.104528 + 0.994522i 0.564602 + 0.251377i 1.49622 + 1.66172i 0.500000 1.53884i −1.51351 + 2.17009i 1.80902 + 1.31433i 1.95630 + 0.415823i −1.80902 3.13331i
25.1 −0.564602 0.251377i −0.978148 0.207912i −1.08268 1.20243i 0.233733 2.22382i 0.500000 + 0.363271i 1.59618 2.11002i 0.690983 + 2.12663i −1.82709 0.813473i −0.690983 + 1.19682i
37.1 −0.564602 + 0.251377i −0.978148 + 0.207912i −1.08268 + 1.20243i 0.233733 + 2.22382i 0.500000 0.363271i 1.59618 + 2.11002i 0.690983 2.12663i −1.82709 + 0.813473i −0.690983 1.19682i
53.1 −1.58268 + 0.336408i −0.104528 0.994522i 0.564602 0.251377i 1.49622 1.66172i 0.500000 + 1.53884i −1.51351 2.17009i 1.80902 1.31433i 1.95630 0.415823i −1.80902 + 3.13331i
58.1 0.0646021 + 0.614648i 0.669131 0.743145i 1.58268 0.336408i −2.04275 + 0.909491i 0.500000 + 0.363271i −0.0510966 2.64526i 0.690983 + 2.12663i 0.209057 + 1.98904i −0.690983 1.19682i
60.1 1.08268 1.20243i 0.913545 0.406737i −0.0646021 0.614648i −2.18720 + 0.464905i 0.500000 1.53884i −2.53158 + 0.768834i 1.80902 + 1.31433i −1.33826 + 1.48629i −1.80902 + 3.13331i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 60.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.m.a 8
3.b odd 2 1 693.2.by.a 8
7.b odd 2 1 539.2.q.a 8
7.c even 3 1 inner 77.2.m.a 8
7.c even 3 1 539.2.f.a 4
7.d odd 6 1 539.2.f.b 4
7.d odd 6 1 539.2.q.a 8
11.b odd 2 1 847.2.n.b 8
11.c even 5 1 inner 77.2.m.a 8
11.c even 5 1 847.2.e.a 4
11.c even 5 2 847.2.n.c 8
11.d odd 10 1 847.2.e.b 4
11.d odd 10 2 847.2.n.a 8
11.d odd 10 1 847.2.n.b 8
21.h odd 6 1 693.2.by.a 8
33.h odd 10 1 693.2.by.a 8
77.h odd 6 1 847.2.n.b 8
77.j odd 10 1 539.2.q.a 8
77.m even 15 1 inner 77.2.m.a 8
77.m even 15 1 539.2.f.a 4
77.m even 15 1 847.2.e.a 4
77.m even 15 2 847.2.n.c 8
77.m even 15 1 5929.2.a.q 2
77.n even 30 1 5929.2.a.j 2
77.o odd 30 1 847.2.e.b 4
77.o odd 30 2 847.2.n.a 8
77.o odd 30 1 847.2.n.b 8
77.o odd 30 1 5929.2.a.l 2
77.p odd 30 1 539.2.f.b 4
77.p odd 30 1 539.2.q.a 8
77.p odd 30 1 5929.2.a.o 2
231.z odd 30 1 693.2.by.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.a 8 1.a even 1 1 trivial
77.2.m.a 8 7.c even 3 1 inner
77.2.m.a 8 11.c even 5 1 inner
77.2.m.a 8 77.m even 15 1 inner
539.2.f.a 4 7.c even 3 1
539.2.f.a 4 77.m even 15 1
539.2.f.b 4 7.d odd 6 1
539.2.f.b 4 77.p odd 30 1
539.2.q.a 8 7.b odd 2 1
539.2.q.a 8 7.d odd 6 1
539.2.q.a 8 77.j odd 10 1
539.2.q.a 8 77.p odd 30 1
693.2.by.a 8 3.b odd 2 1
693.2.by.a 8 21.h odd 6 1
693.2.by.a 8 33.h odd 10 1
693.2.by.a 8 231.z odd 30 1
847.2.e.a 4 11.c even 5 1
847.2.e.a 4 77.m even 15 1
847.2.e.b 4 11.d odd 10 1
847.2.e.b 4 77.o odd 30 1
847.2.n.a 8 11.d odd 10 2
847.2.n.a 8 77.o odd 30 2
847.2.n.b 8 11.b odd 2 1
847.2.n.b 8 11.d odd 10 1
847.2.n.b 8 77.h odd 6 1
847.2.n.b 8 77.o odd 30 1
847.2.n.c 8 11.c even 5 2
847.2.n.c 8 77.m even 15 2
5929.2.a.j 2 77.n even 30 1
5929.2.a.l 2 77.o odd 30 1
5929.2.a.o 2 77.p odd 30 1
5929.2.a.q 2 77.m even 15 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 2 T_{2}^{7} + 2 T_{2}^{5} + 9 T_{2}^{4} + 8 T_{2}^{3} + 5 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(77, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} - 6 T^{3} - 17 T^{4} - 24 T^{5} - T^{6} + 47 T^{7} + 103 T^{8} + 94 T^{9} - 4 T^{10} - 192 T^{11} - 272 T^{12} - 192 T^{13} + 128 T^{14} + 256 T^{15} + 256 T^{16}$$
$3$ $$1 - T + 3 T^{2} - 8 T^{3} + 8 T^{4} + 7 T^{5} + 6 T^{6} + 56 T^{7} - 137 T^{8} + 168 T^{9} + 54 T^{10} + 189 T^{11} + 648 T^{12} - 1944 T^{13} + 2187 T^{14} - 2187 T^{15} + 6561 T^{16}$$
$5$ $$( 1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4} )^{2}( 1 - 5 T + 10 T^{2} - 25 T^{3} + 75 T^{4} - 125 T^{5} + 250 T^{6} - 625 T^{7} + 625 T^{8} )$$
$7$ $$1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8}$$
$11$ $$1 - 4 T + 10 T^{2} + 64 T^{3} - 261 T^{4} + 704 T^{5} + 1210 T^{6} - 5324 T^{7} + 14641 T^{8}$$
$13$ $$( 1 + 5 T + 27 T^{2} + 115 T^{3} + 584 T^{4} + 1495 T^{5} + 4563 T^{6} + 10985 T^{7} + 28561 T^{8} )^{2}$$
$17$ $$1 + 4 T + 17 T^{2} - 120 T^{3} - 740 T^{4} - 3144 T^{5} - 649 T^{6} + 49990 T^{7} + 275299 T^{8} + 849830 T^{9} - 187561 T^{10} - 15446472 T^{11} - 61805540 T^{12} - 170382840 T^{13} + 410338673 T^{14} + 1641354692 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 3 T - 26 T^{2} - 21 T^{3} + 252 T^{4} - 1302 T^{5} - 1792 T^{6} + 24084 T^{7} + 90161 T^{8} + 457596 T^{9} - 646912 T^{10} - 8930418 T^{11} + 32840892 T^{12} - 51998079 T^{13} - 1223192906 T^{14} + 2681615217 T^{15} + 16983563041 T^{16}$$
$23$ $$( 1 - 8 T + 7 T^{2} - 88 T^{3} + 1248 T^{4} - 2024 T^{5} + 3703 T^{6} - 97336 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 12 T + 25 T^{2} + 288 T^{3} - 2471 T^{4} + 8352 T^{5} + 21025 T^{6} - 292668 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 19 T + 210 T^{2} - 1691 T^{3} + 10649 T^{4} - 52421 T^{5} + 201810 T^{6} - 566029 T^{7} + 923521 T^{8} )( 1 + 11 T + 60 T^{2} - 11 T^{3} - 991 T^{4} - 341 T^{5} + 57660 T^{6} + 327701 T^{7} + 923521 T^{8} )$$
$37$ $$1 + 13 T + 112 T^{2} + 71 T^{3} - 4102 T^{4} - 46406 T^{5} - 63616 T^{6} + 1263548 T^{7} + 15622363 T^{8} + 46751276 T^{9} - 87090304 T^{10} - 2350603118 T^{11} - 7687808422 T^{12} + 4923420947 T^{13} + 287361357808 T^{14} + 1234114402729 T^{15} + 3512479453921 T^{16}$$
$41$ $$( 1 - T - 25 T^{2} + 221 T^{3} + 1064 T^{4} + 9061 T^{5} - 42025 T^{6} - 68921 T^{7} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 7 T + 87 T^{2} - 301 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 - 6 T + 7 T^{2} + 690 T^{3} - 6420 T^{4} + 23196 T^{5} + 57521 T^{6} - 1846680 T^{7} + 13766039 T^{8} - 86793960 T^{9} + 127063889 T^{10} + 2408278308 T^{11} - 31327552020 T^{12} + 158248054830 T^{13} + 75454507303 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 12 T + 103 T^{2} - 420 T^{3} - 9840 T^{4} - 102312 T^{5} - 99751 T^{6} + 3945120 T^{7} + 55833959 T^{8} + 209091360 T^{9} - 280200559 T^{10} - 15231903624 T^{11} - 77642333040 T^{12} - 175642107060 T^{13} + 2282929196287 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 18 T + 239 T^{2} + 1374 T^{3} + 5292 T^{4} - 7812 T^{5} + 470053 T^{6} + 8281674 T^{7} + 95738891 T^{8} + 488618766 T^{9} + 1636254493 T^{10} - 1604420748 T^{11} + 64125074412 T^{12} + 982305986826 T^{13} + 10081147540199 T^{14} + 44795726726742 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 18 T + 141 T^{2} + 538 T^{3} - 19524 T^{4} + 199396 T^{5} - 460901 T^{6} - 9059916 T^{7} + 127866487 T^{8} - 552654876 T^{9} - 1715012621 T^{10} + 45259103476 T^{11} - 270326199684 T^{12} + 454392809938 T^{13} + 7264372784901 T^{14} - 56569371048378 T^{15} + 191707312997281 T^{16}$$
$67$ $$( 1 + 19 T + 138 T^{2} + 1691 T^{3} + 21053 T^{4} + 113297 T^{5} + 619482 T^{6} + 5714497 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 30814 T^{5} - 236927 T^{6} + 2863288 T^{7} + 25411681 T^{8} )^{2}$$
$73$ $$1 - 15 T + 208 T^{2} - 2445 T^{3} + 29070 T^{4} - 272130 T^{5} + 2789288 T^{6} - 23441760 T^{7} + 210261239 T^{8} - 1711248480 T^{9} + 14864115752 T^{10} - 105863196210 T^{11} + 825536865870 T^{12} - 5068660044885 T^{13} + 31477519068112 T^{14} - 165710977786455 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 9 T - 11 T^{2} + 1446 T^{3} - 17334 T^{4} + 85743 T^{5} + 200726 T^{6} - 10774872 T^{7} + 128238887 T^{8} - 851214888 T^{9} + 1252730966 T^{10} + 42274642977 T^{11} - 675160704054 T^{12} + 4449423552954 T^{13} - 2673962010731 T^{14} - 172835180875431 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 - 9 T - 2 T^{2} + 765 T^{3} - 6719 T^{4} + 63495 T^{5} - 13778 T^{6} - 5146083 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 12 T - 50 T^{2} + 192 T^{3} + 16899 T^{4} + 17088 T^{5} - 396050 T^{6} + 8459628 T^{7} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 7 T - 63 T^{2} + 185 T^{3} + 11276 T^{4} + 17945 T^{5} - 592767 T^{6} + 6388711 T^{7} + 88529281 T^{8} )^{2}$$